The equivalence between these two descriptions is best understood
by moving from set-valued to groupoid-valued deformation functors;
see, e.g., [1]}, [2]}. Associating with any open set
\(U\)  in \(X\)  the groupoid \(\operatorname{Def}_{\mathcal {E}|_U}\)  of infinitesimal
deformations of \(\mathcal {E}\)  over \(U\)  (over a fixed base \(\operatorname{Spec}A\) , for
some local Artin ring \(A\) ) defines a stack over \({\bf {Top}}_X\) ; this is
just a one-word way of saying that global deformations of \(\mathcal {E}\) 
are the same thing as the descent data for its local deformations:
\( \operatorname{Def}_\mathcal {E}\simeq \displaystyle \mathop {\rm holim}_{U\in \Delta _\mathcal {U}} \operatorname{Def}_{\mathcal {E}|_U}, \) 
